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   Research Article

Dynamical Transitions in a Pollination-Herbivory Interaction: A Conflict between
Mutualism and Antagonism

     * Tomás A. Revilla ,
       * E-mail: [50]tomrevilla@gmail.com
       Current address: Biology Centre, Academy of Sciences of the Czech Republic,
       Ceské Budejovice, Czech Republic
       Affiliation Centre for Biodiversity Theory and Modelling, Station d'Ecologie
       Expérimentale du Centre National de la Recherche Scientifique à Moulis,
       Moulis, France
       x
     * Francisco Encinas-Viso
       Current address: Centre for Australian National Biodiversity Research,
       Commonwealth Scientific and Industrial Research Organisation, National
       Facilities and Collections, Canberra, Australia
       Affiliation Community and Conservation Ecology Group, Centre for Ecological
       and Evolutionary Studies, University of Groningen, Groningen, The Netherlands
       x

Dynamical Transitions in a Pollination-Herbivory Interaction: A Conflict between
Mutualism and Antagonism

     * Tomás A. Revilla,
     * Francisco Encinas-Viso

   PLOS
   x
     * Published: February 20, 2015
     * [51]https://doi.org/10.1371/journal.pone.0117964
     *

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     * [57]Figures

Figures

   Fig 1
   Table 1
   Fig 2
   Fig 3
   Fig 4
   Fig 5

Abstract

   Plant-pollinator associations are often seen as purely mutualistic, while in
   reality they can be more complex. Indeed they may also display a diverse array of
   antagonistic interactions, such as competition and victim-exploiter interactions.
   In some cases mutualistic and antagonistic interactions are carried-out by the
   same species but at different life-stages. As a consequence, population structure
   affects the balance of inter-specific associations, a topic that is receiving
   increased attention. In this paper, we developed a model that captures the basic
   features of the interaction between a flowering plant and an insect with a larval
   stage that feeds on the plant's vegetative tissues (e.g. leaves) and an adult
   pollinator stage. Our model is able to display a rich set of dynamics, the most
   remarkable of which involves victim-exploiter oscillations that allow plants to
   attain abundances above their carrying capacities and the periodic alternation
   between states dominated by mutualism or antagonism. Our study indicates that
   changes in the insect's life cycle can modify the balance between mutualism and
   antagonism, causing important qualitative changes in the interaction dynamics.
   These changes in the life cycle could be caused by a variety of external drivers,
   such as temperature, plant nutrients, pesticides and changes in the diet of adult
   pollinators.

   Citation: Revilla TA, Encinas-Viso F (2015) Dynamical Transitions in a
   Pollination-Herbivory Interaction: A Conflict between Mutualism and Antagonism.
   PLoS ONE 10(2): e0117964. https://doi.org/10.1371/journal.pone.0117964

   Academic Editor: Jordi Garcia-Ojalvo, Universitat Pompeu Fabra, SPAIN

   Received: October 14, 2014; Accepted: January 6, 2015; Published: February 20,
   2015

   Copyright: © 2015 Revilla, Encinas-Viso. This is an open access article
   distributed under the terms of the [58]Creative Commons Attribution License,
   which permits unrestricted use, distribution, and reproduction in any medium,
   provided the original author and source are credited

   Data Availability: All relevant data consists of graphs generated by computer
   scripts provided in the supplementary information file.

   Funding: TAR was supported by the TULIP Laboratory of Excellence
   (ANR-10-LABX-41). FEV was supported by the OCE postdoctoral fellowship at CSIRO.
   The funders had no role in study design, data collection and analysis, decision
   to publish, or preparation of the manuscript.

   Competing interests: The authors have declared that no competing interests exist.

Introduction

     Il faut bien que je supporte deux ou trois chenilles si je veux connaître les
     papillons

     Le Petit Prince, Chapitre IX - Antoine de Saint-Exupéry

   Mutualism can be broadly defined as cooperation between different species
   [[59]1]. In mutualistic interactions typically there are benefits and costs, in
   terms of resources, energy and time devoted to them, but the net outcome is (+,+)
   in the final balance. However, there can be other kinds of costs, concerning
   detrimental interactions that run in parallel with mutualism, such as predation,
   parasitism or competition, involving the same parties. Moreover, some of these
   antagonistic interactions (e.g. competition) seem to be important for the
   evolution and stability of mutualism [[60]2]. In general, these costs have
   important consequences at the population and community level because the net
   outcome of an interspecific association can turn out beneficial or detrimental
   and more interestingly, variable [[61]3]. Variable interactions challenge the
   view that ecological communities are structured by well defined interactions at
   the species level such as competition (-,-), victim-exploiter (-,+) or mutualism
   (+,+).

   Pollination is one of the most important mutualisms occurring between plants and
   animals. This form of trading resources for services greatly explains the
   evolutionary success of flowering plants in almost all terrestrial systems. It is
   responsible for the well being of ecosystem services. During the larval stage of
   many insect pollinators, such as Lepidopterans (butterflies and moths), the
   larvae feed on plant leaves to mature and become adult pollinators [[62]4-[63]7].
   These ontogenetic diet shifts [[64]8] are very common and important in
   understanding the ecological and evolutionary dynamics of plant-animal
   mutualisms. Interestingly, in some cases larvae feed on the same plant species
   that they will pollinate as adults [[65]6, [66]9]. This shows that in several
   cases mutualistic and antagonistic interactions are exerted by the same species,
   and a potential conflict arises for the plant, between the benefits of mutualism
   and the costs of herbivory. One of the best known examples is the interaction
   between tobacco plants (Nicotiana attenuata) and the hawkmoth (Manduca sexta)
   [[67]10, [68]11], whose larva is commonly called the tobacco hornworm. There are
   other examples of this type of interaction in the genus Manduca (Sphingidae),
   such as between the tomato plant (Lycopersicon esculentum) and the five-spotted
   hawkmoth (Manduca quinquemaculata) [[69]12]. These larvae have received a lot of
   attention due to their negative effects on agricultural crops [[70]13].

   The interaction between Manduca sexta and Datura wrightii (Solanacea) [[71]6,
   [72]14] is another good example illustrating the costs and benefits of
   pollination mutualisms [[73]6]. D. wrightii provides high volumes of nectar and
   seems to depend heavily on the pollination service by M. sexta adults [[74]14].
   However, M. sexta larvae, which feed on D. wrightii vegetative tissue, can have
   severe negative effects on plant fitness [[75]15, [76]16]. We could assume that
   the benefits of pollination might outweigh the costs of herbivory for this
   mutualism to be relatively viable. The question is what are the conditions, in
   terms of benefits (pollination) and costs (herbivory), for this mutualistic
   interaction to be stable?

   In the pollination-herbivory cases mentioned previously the benefits and costs
   for the plant are clearly differentiated. This is because the role of an insect
   as a pollinator or herbivore depends on the stage in its life cycle [[77]17].
   Thus, whether mutualism or herbivory dominates the interaction is dependent on
   insect abundance and its population structure. In other words the cost:benefit
   ratio must be positively related with the insect's larva:adult ratio. For a
   hypothetical scenario in which the costs of herbivory (-) and the benefits of
   pollination (+) are balanced for the plant (0), an increase in larval abundance
   relative to adults should bias the relationship towards a victim-exploiter one
   (-,+). Whereas an increase in adult abundance relative to larvae should bias the
   relationship towards mutualism (+,+). Under equilibrium conditions, one would
   expect transitions (bifurcations) from (-,+) to (0,+) to (+,+) and vice-versa as
   relevant parameters affecting the plant and the insect life-histories vary, such
   as flower production, mortalities or larvae maturation rates. However, under
   dynamic scenarios the outcome may be more complex: a victim-exploiter state (-,+)
   enhances larva development into pollinating adults, but this tips the interaction
   into a mutualism (+,+), which in turn contributes greater production of larva
   leading back to a victim-exploiter state (-,+). This raises the possibility of
   feedback between the plant-insect interaction and insect population structure,
   which can potentially lead to periodic alternation between mutualism and
   herbivory. Thus, when non-equilibrium dynamics are involved, questions concerning
   the overall nature (positive, neutral or negative) of mixed interactions may not
   have simple answers.

   In this article we study the feedback between insect population structure,
   pollination and herbivory. We want to understand how the balance between costs
   (herbivory) and benefits (pollination) affects the interaction between plants
   (e.g. D. wrightii) and herbivore-pollinator insects (e.g. M. sexta)? Also what
   role does insect development have in this balance and on the resulting dynamics?
   We use a mathematical model which considers two different resources provided by
   the same plant species, nectar and vegetative tissues. Nectar consumption
   benefits the plant in the form of fertilized ovules, and consumption of
   vegetative tissues by larvae causes a cost. Our model predicts that the balance
   between mutualism and antagonism, and the long term stability of the plant-insect
   association, can be greatly affected by changes in larval development rates, as
   well as by changes in the diet of adult pollinators.

Methods

   Our model concerns the dynamics of the interaction between a plant and an insect.
   The insect life cycle comprises an adult phase that pollinates the flowers and a
   larval phase that feed on non-reproductive tissues of the same plant. Adults
   oviposit on the same species that they pollinate (e.g. D. wrightii - M. sexta
   interaction). Let denote the biomass densities of the plant, the larva, and the
   adult insect with P, L and A respectively. An additional variable, the total
   biomass of flowers F, enables the mutualism by providing resources to the insect
   (nectar), and by collecting services for the plant (pollination). The
   relationship is facultative-obligatory. In the absence of pollination, plant
   biomass persists by vegetative growth (e.g. root, stem and leaf biomass are being
   constantly renewed). For the sake of simplicity and because we want to focus on
   the plant-insect interaction, we describe vegetative growth using a logistic
   growth rate, a choice that is empirically justified for tobacco plants [[78]18].
   In the absence of the plant, however, the insect always goes extinct because
   larval development relies exclusively on herbivory, even if adults pollinate
   other plant species. This is based on the biology of M. sexta [[79]6]. The
   mechanism of interaction between these four variables (P, L, A, F), as shown in
   [80]Fig. 1, is described by the following system of ordinary differential
   equations (ODE): [journal.pone.0117964.e001] (1) where r: plant intrinsic growth
   rate, c: plant intra-specific self-regulation coefficient (also the inverse its
   carrying capacity), a: pollination rate, b: herbivory rate, s: flower production
   rate, w: flower decay rate, m, n: larva and adult mortality rates, s: plant
   pollination efficiency ratio, e: adult consumption efficiency ratio. Like e,
   parameter g is also a consumption efficiency ratio, but we will call it the
   maturation rate for brevity since we will refer to it frequently. Our model
   assumes that pollination leads to flower closure [[81]19], causing resource
   limitation for adult insects. Parameter g represents a reproduction rate
   resulting from the pollination of other plants species, which we do not model
   explicitly. Most of our results are for g = 0.
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   Fig 1. Interaction mechanism between plants (P), flowers (F), larva (L), adult
   insects (A) and associated biomass flows.

   Clipart sources: [83]http://etc.usf.edu/clipart/

   [84]https://doi.org/10.1371/journal.pone.0117964.g001

   We now consider the fact that flowers are ephemeral compared with the life cycles
   of plants and insects. In other words, some variables (P, L, A) have slower
   dynamics, and others (F) are fast [[85]20]. Given the near constancy of plants
   and animals in the flower equation of ([86]1), we can predict that flowers will
   approach a quasi-steady-state (or quasi-equilibrium) biomass F ~= sP/(w + aA),
   before P, L and A can vary appreciably. Substituting the quasi-steady-state
   biomass in system ([87]1) we arrive at: [journal.pone.0117964.e002] (2)

   In system ([88]2) the quantities in square brackets can be regarded as functional
   responses. Plant benefits saturate with adult pollinator biomass, i.e.
   pollination exhibits diminishing returns. The functional response for the insects
   is linear in the plant biomass, but is affected by intraspecific competition
   [[89]21] for mutualistic resources.

   We non-dimensionalized this model to reduce the parameter space from 12 to 9
   parameters, by casting biomasses with respect to the plant's carrying capacity
   (1/c) and time in units of plant biomass renewal time (1/r). This results in a
   PLA (plant, larva, adult) scaled model: [journal.pone.0117964.e003] (3)

   [90]Table 1 lists the relevant transformations.
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   Table 1. Variables and parameters.

   [92]https://doi.org/10.1371/journal.pone.0117964.t001

   There is an important clarification to make concerning the nature and scales of
   the conversion efficiency ratios s, e involved in pollination, and g for
   herbivory and maturation. This has to do with the fact that flowers per se are
   not resources or services, but organs that enable the mutualism to take place,
   and they mean different things in terms of biomass production for plants and
   animals. For insects, the yield of pollination is thermodynamically constrained.
   First of all, a given biomass F of flowers contains an amount of nectar that is
   necessarily less than F. More importantly, part of this nectar is devoted to
   survival, or wasted, leaving even less for reproduction. Similarly, not all the
   biomass consumed by larvae will contribute to their maturation to adult. Ergo e <
   1, g < 1. Regarding the returns from pollination for the plants, the situation is
   very different. Each flower harbors a large number of ovules, thus a potentially
   large number of seeds [[93]22], each of which will increase in biomass by
   consuming resources not considered by our model (e.g. nutrients, light).
   Consequently, a given biomass of pollinated flowers can produce a larger biomass
   of mature plants, making s larger than 1.

   The PLA model ([94]3) has many parameters. However, here we focus on herbivory
   rates (b) and larvae maturation (g) because increasing b turns the net balance
   interaction towards antagonism, whereas increasing g shifts insect population
   structure towards the adult phase, turning the net balance towards mutualism.
   Both parameters also relate to the state variables at equilibrium (i.e. z/y =
   bgx/n in ([95]3) for dz/dt = 0). We studied the joint effects of varying b and g
   numerically (parameter values in [96]Table 1) using XPPAUT [[97]23]. ODE were
   integrated using Matlab [[98]24] or GNU/Octave [[99]25]. We also present a
   simplified graphical analysis of our model, in order to explain how different
   dynamics can arise, by varying other parameters. The source codes supporting
   these results are provided as supplementary material ([100]S1 File).

Results

Numerical results

   [101]Fig. 2 shows interaction outcomes of the PLA model, as a function of b and g
   for specialist pollinators (phi = 0). This parameter space is divided by a
   decreasing R[o] = 1 line that indicates whether or not insects can invade when
   rare. R[o] is defined as (see derivation in [102]S1 File):
   [journal.pone.0117964.e004] (4) and we call it the basic reproductive number,
   according to the argument that follows. Consider the following in system
   ([103]3): if the plant is at carrying capacity (x = 1), and is invaded by a very
   small number of adult insects (z ~= 0), the average number of larvae produced by
   a single adult in a given instant is eax/(y+z) ~= ea/y, and during its life-time
   (n^-1) it is ea/yn. Larvae die at the rate µ, or mature with a rate equal to gbx
   = gb, per larva. Thus, the probability of larvae becoming adults rather than
   dying is gb/(µ+gb). Multiplying the life-time contribution of an adult by this
   probability gives the expected number of new adults replacing one adult per
   generation during an invasion (R[o]). More formally, R[o] is the expected number
   of adult-insect-grams replacing one adult-insect-gram per generation (assuming a
   constant mass-per-individual ratio).
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   Fig 2. Outcomes of the PLA model as a function of the larval maturation and
   herbivory rates for specialist pollinators (phi = 0).

   The rectangular region in the bottom left is analyzed with more detail in [105]S1
   File.

   [106]https://doi.org/10.1371/journal.pone.0117964.g002

   Below the R[o] = 1 line, small insect populations cannot replace themselves (R[o]
   < 1) and two outcomes are possible. If the maturation rate is too low, the plant
   only equilibrium (x = 1, y = z = 0) is globally stable and plant-insect
   coexistence is impossible for all initial conditions. If the maturation rate is
   large enough, stable coexistence is possible, but only if the initial plant and
   insect biomass are large enough. This is expected in models where at least one
   species, here the insect, is an obligate mutualist. In this region of the space
   of parameters, the growth of small insect populations increases with population
   size, a phenomenon called the Allee effect [[107]26].

   Above the R[o] = 1 line the plant only equilibrium is always unstable against the
   invasion of small insect populations (R[o] > 1). Plants and insects can coexist
   in a stable equilibrium or via limit cycles (stable oscillations). The zone of
   limit cycles occurs for intermediate values of the maturation rate (g) and it
   widens with rate of herbivory (b).

   Plant equilibrium when coexisting with insects can be above or below the carrying
   capacity (x = 1). When above carrying capacity the net result of the interaction
   is a mutualism (+,+). While in the second case we have antagonism, more
   specifically net herbivory (-,+). As it would be expected, increasing herbivory
   rates (b) shifts this net balance towards antagonism (low plant biomass), while
   decreasing it shifts the balance towards mutualism (high plant biomass). The
   quantitative response to increases in the maturation rate (g) is more complex
   however (see the bifurcation plot in [108]S1 File).

   Given that there is herbivory, we encounter victim-exploiter oscillations.
   However, the oscillations in the PLA model are special in the sense that the
   plant can attain maximum biomasses above the carrying capacity (x > 1). For an
   example see [109]Fig. 3. Instead of a stable balance between antagonism and
   mutualism, we can say that the outcome in [110]Fig. 3 is a periodic alternation
   of both cases. This is not seen in simple victim-exploiter models, where
   oscillations are always below the victim's carrying capacity [[111]27, [112]28].
   The relative position of the cycles along the plant axis is also affected by
   herbivory: if b decreases (increases), plant maxima and minima will increase
   (decrease) in [113]Fig. 3 (see bifurcation plot in [114]S1 File). In some cases
   the entire plant cycle (maxima and minima) ends above the carrying capacity if b
   is low enough (see [115]S1 File), but further decrease causes damped
   oscillations. We also found examples in which coexistence can be stable or lead
   to limit cycles depending on the initial conditions (see example in [116]S1
   File), but this happens in a very restrictive region in the space of parameters
   (see bifurcation plot in [117]S1 File). Limit cycles can also cross the plant's
   carrying capacity under the original interaction mechanism ([118]1), which does
   not assume the steady-state in the flowers (see [119]S1 File, using parameters in
   the last column of [120]Table 1).
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   Fig 3. Limit cycles in the PLA model ([122]3).

   Plant biomass alternates above and below the carrying capacity (dotted line).
   Parameters as in [123]Table 1, with g = 0.01, b = 10. Blue:plant, green:larva,
   red:adult.

   [124]https://doi.org/10.1371/journal.pone.0117964.g003

   [125]Fig. 4 shows the b vs g parameter space of the model when the adults are
   more generalist. The relative positions of the plant-only, Allee effect, and
   coexistence regions are similar to the case of specialist pollinators ([126]Fig.
   2). However, the region of limit cycles is much larger. The R[0] = 1 line is
   closer to the origin, because the expression for R[0] is now (see derivation in
   [127]S1 File): [journal.pone.0117964.e005] (5)
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   Fig 4. Outcomes of the PLA model as a function of the larval maturation and
   herbivory rates for generalist pollinators (phi = 1).

   AeLc: intersection of the Allee effect and Limit cycle zones.

   [129]https://doi.org/10.1371/journal.pone.0117964.g004

   In other words, this means that the more generalist the adult pollinators (larger
   phi ), the more likely they can invade when rare. There is also a small overlap
   between the Allee effect and limit cycle regions, i.e. parameter combinations for
   which the long term outcome could be insect extinction or plant-insect
   oscillations, depending on the initial conditions.

Graphical analysis

   The general features of the interaction can be studied by phase-plane analysis.
   To make this easier, we collapsed the three-dimensional PLA model into a
   two-dimensional plant-larva (PL) model, by assuming that adults are extremely
   short lived compared with plants and larvae (see resulting ODE in [130]S1 File).
   The closest realization of this assumption could be Manduca sexta, which has a
   larval stage of approximately 20-25 days and adult stages of around 7 days
   [[131]29, [132]30]. For a given parametrization ([133]Table 1), the PL model has
   the same equilibria as the PLA model, but not the exact same global dynamics due
   to the alteration of time scales. Yet, this simplification provides insights
   about the outcomes displayed in Figs. [134]2 and [135]4.

   [136]Fig. 5 shows representative examples of plant and larva isoclines (i.e.
   non-trivial nullclines) and coexistence equilibria (intersections). Isocline
   properties are analytically justified (see [137]S1 File and supplemented
   [[138]31] worksheet). The local dynamics around equilibria depends on the
   eigenvalues of the jacobian matrix of the PL model at the equilibrium. However,
   the highly non-linear nature of the PL model (see [139]S1 File), makes it
   pointless to try infer the signs of the eigenvalues by analytical means (except
   for trivial and plant-only equilibrium). Thus, we propose to use to local
   geometry of isocline intersections to infer local stability [[140]32]. Plant
   isoclines take two main forms: [journal.pone.0117964.e006] (6) In both cases,
   plants grow between the isocline and the axes, and decrease otherwise. Larva
   isoclines are simpler, they start in the plant axis and bend towards the right
   when insects tend towards specialization (phi < n), as shown by [141]Fig. 5. When
   insects tend towards generalism (phi > n), their isoclines increase rapidly
   upwards like the letter "J" (not shown here, see [142]S1 File). Insects grow
   below and right of the larva isocline, and decrease otherwise.
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   Fig 5. Dynamics of the simplified version of the PLA model.

   Plant isoclines in green and larva isoclines in blue. Several trajectories are
   shown (starting with *). The dotted line at x = 1 is the plant's carrying
   capacity. When gsa/yn < 1 the plant's isocline always decreases, when gsa/yn > 1,
   it bulges above the carrying capacity and displays a hump. (A) Damped
   oscillations leading to globally stable coexistence dominated by antagonism
   (victim-exploiter). (B) The isoclines intersect as a locally stable mutualistic
   equilibrium and as a saddle point. Insects can coexist with the plant or go
   extinct depending on the initial conditions. (C) This is similar to case (B),
   however, a stable mutualism occurs only after damped oscillations or the insect
   go extinct, depending on the initial conditions. (D) Here the system develops
   oscillations approaching a limit cycle (thick loop), which creates a periodic
   alternation between mutualism and antagonism. Common parameters in all panels are
   b = 10, y = 0.1, µ = 1, phi = 0. For the other parameters; in (A): s = 3, e =
   0.7, a = 3, g = 0.02, n = 2; in (B): s = 2.1, e = 0.21, a = 2, g = 0.05, n = 1.5;
   in (C): s = 3.7, e = 0.2, a = 3, g = 0.02, n = 1.5; in (D): s = 5, e = 0.3, a =
   5, g = 0.02, n = 2.

   [144]https://doi.org/10.1371/journal.pone.0117964.g005

   The gsa < yn case in [145]Fig. 5A illustrates scenarios in which pollination
   rates (a), plant benefits (s), adult pollinator lifetimes (1/n) and
   larva-to-adult transition rates (g) are low. The plant's isocline is a decreasing
   curve crossing the plant's axis at its carrying capacity K (x = 1, y = 0). The
   intersection with the larva isocline creates a globally stable equilibrium,
   approached by oscillations of decreasing amplitude. The local stability of this
   equilibrium can be explained partly by the geometry of the intersection:
   [146]Fig. 5A shows that if plants increase (decrease) above (below) the
   intersection point, while keeping the insect density fixed, they enter a zone of
   negative (positive) growth; and the same behavior holds for the insects while
   keeping the plants fixed. In ecological terms, both species are self-limited
   around the equilibrium, a strong indication of stability [[147]32]. Together with
   the fact that the trivial (x = 0, y = 0) and carrying capacity equilibrium (x =
   1, y = 0) are saddle points, we conclude that plants and insects achieve a
   globally stable equilibrium after a period of transient oscillations (provided
   that insects are viable, e.g. b, g, e are large enough). This equilibrium is
   demographically unfavorable for the plant because its biomass lies below the
   carrying capacity (x < 1). Indeed, for extreme scenarios of negligible plant
   pollination benefits (i.e. a and/or s tend to zero), the plant's isocline
   approximates a straight line with a negative slope, like the isocline of a
   logistic prey in a Lotka-Volterra model, which is well known to cause damped
   oscillations [[148]32].

   The gsa > yn case in [149]Figs. 5B,C,D cover scenarios in which pollination rates
   (a), pollination benefits (s), adult pollinator lifetimes (1/n) and
   larva-to-adult (harm-to-benefit) transition rates (g) are high. One part of the
   plant's isocline lies above the carrying capacity, which means that coexistence
   equilibria with plant biomass larger than the carrying capacity (x > 1) are
   possible, and this is favorable for the plant. [150]Fig. 5B, shows and example
   where the larva isocline intersects the plant's isocline twice above the carrying
   capacity. One intersection is a locally stable coexistence equilibrium, whereas
   the other intersection is a saddle point. The saddle point belongs to a boundary
   that separates regions of initial conditions leading to insect persistence or
   extinction. This can explain the Allee effect, i.e. insect growth rates increase
   (go from negative to positive) with insect density when insect populations are
   very small.

   As the second inequality of ([151]6) widens (gsa >> yn), the plant's isocline
   takes a mushroom-like shape (or "anvil" or letter "W*"), as in [152]Fig. 5C,D.
   The plant's isocline displays a very prominent "hump", like in the prey isocline
   of the Rosenzweig-MacArthur model [[153]27]. As a "rule of thumb", intersections
   at the right of the hump would lead to damped oscillations, for the reasons
   explained before ([154]Fig. 5A, for gsa < yn). Also as a "rule of thumb",
   intersections at the left of the hump (like in [155]Fig. 5C,D) are expected to
   result in reduced stability. This is because a small increase (decrease) along
   the plant's axis leaves the plant at the growing (decreasing) side of its
   isocline, promoting further increase (decrease). This means that plants do not
   experience self-limitation, which is an indication of instability [[156]32], and
   we infer that oscillations will not vanish. [157]Fig. 5D shows an example where
   an intersection at the left of the hump causes instability, leading to limit
   cycles. However, [158]Fig. 5C shows an exception of this prediction (the
   intersection is stable). In both examples the intersection occurs above the
   plants carrying capacity, thus revealing oscillations alternating above and below
   the plant's carrying capacity. We want to stress one more time, that these
   predictions based on isocline intersection configurations (left vs right of the
   hump) must be taken as "rules of thumb".

   [159]Fig. 5C also reveals an important consequence of the dual interaction
   between the plant and the insect. As we can see, the presence of a saddle point
   leads to the Allee effect explained before. But this figure also shows that large
   larval densities can lead to insect extinction. This can be explained by the fact
   that at large initial densities, the larva overexploits the plant, and this is
   followed by an insect population crash from which it cannot recover due to the
   Allee effect.

   As g, s, a increase and/or y, n decrease more and more, the decreasing segment of
   the plant isocline (the part at the right of the hump) approximates a decreasing
   line (actually a straight asymptotic line, see [160]S1 File), while the rest of
   the isocline is pushed closer and closer to the axes. In other words, when
   pollination rates (a), benefits (s), adult lifetimes (1/n) and larva development
   rates (g) increase, plant isoclines would resemble the isocline of a logistic
   prey, with a "pseudo" carrying capacity (the rightmost extent of the isocline)
   larger than the intrinsic carrying capacity (x = 1). [161]Fig. 5D is an example
   of this. These conditions would promote stable coexistence with large plant
   equilibrium biomasses.

Discussion

   We developed a plant-insect model that considers two interaction types,
   pollination and herbivory. Ours belongs to a class of models [[162]33, [163]34]
   in which balances between costs and benefits cause continuous variation in
   interaction strengths, as well as transitions among interaction types (mutualism,
   predation, competition). In our particular case, interaction types depend on the
   stage of the insect's life cycle, as inspired by the interaction between M. sexta
   and D. wrightii [[164]6, [165]14] or between M. sexta and N. attenuata [[166]10].
   There are many other examples of pollination-herbivory in Lepidopterans, where
   adult butterflies pollinate the same plants exploited by their larvae [[167]5,
   [168]7]. We assign antagonistic and mutualistic roles to larva and adult insect
   stages respectively, which enable us to study the consequences of ontogenetic
   changes on the dynamics of plant-insect associations, a topic that is receiving
   increased attention [[169]8, [170]17]. Our model could be generalized to other
   scenarios, in which drastic ontogenetic niche shifts cause the separation of
   benefits and costs in time and space. However, it excludes cases like the
   yucca/yucca moth interaction [[171]35] where adult pollinated ovules face larval
   predation, i.e. benefits themselves are deducted.

   Instead of using species biomasses as resource and service proxies [[172]34], we
   consider a mechanism ([173]1) that treats resources more explicitly [[174]36]. We
   use flowers as a direct proxy of resource availability, by assuming a uniform
   volume of nectar per flower. Nectar consumption by insects is concomitant with
   service exploitation by the plants (pollination), based on the assumption that
   flowers contain uniform numbers of ovules. Pollination also leads to flower
   closure [[175]19], making them limiting resources. Flowers are ephemeral compared
   with plants and insects, so we consider that they attain a steady-state between
   production and disappearance. As a result, the dynamics is stated only in terms
   of plant, larva and adult populations, i.e. the PLA model ([176]3). The
   feasibility of the results described by our analysis depends on several
   parameters. The consumption, mortalities and growth rates, and the carrying
   capacities (e.g. a, b, m, n and r, c in the fourth column of [177]Table 1), have
   values close to the ranges considered by other models [[178]34, [179]37].
   Oscillations, for example, require large herbivory rates, but this is usual for
   M. sexta [[180]15].

Mutualism-antagonism cycles

   The PLA model displays plant-insect coexistence for any combination of
   (non-trivial) initial conditions where insects can invade when rare (R[o] > 1).
   Coexistence is also possible where insects cannot invade when rare (R[o] < 1),
   but this requires high initial biomasses of plants and insects (Allee effect).
   Coexistence can take the form of a stable equilibrium, but it can also take the
   form of stable oscillations, i.e. limit cycles.

   Previous models combining mutualism and antagonism predict oscillations, but they
   are transient ones [[181]35, [182]38], or the limit cycles occur entirely below
   the plant's carrying capacity [[183]39]. We have good reasons to conclude that
   the cycles are herbivory driven and not simply a consequence of the PLA model
   having many variables and non-linearities. First of all, limit cycles require
   herbivory rates (b) to be large enough. Second, given limit cycles, an increase
   in the maturation rate (g) causes a transition to stable coexistence, and further
   increase in herbivory is required to induce limit cycles again ([184]Fig. 2).
   This makes sense because by speeding up the transition from larva to adult, the
   total effect of herbivory on the plants is reduced, hence preventing a crash in
   plant biomass followed by a crash in the insects. Third, when adult pollinators
   have alternative food sources (phi > 1), the zone of limit cycles in the space of
   parameters becomes larger ([185]Fig. 4). This also makes sense, because the total
   effect of herbivory increases by an additional supply of larva (which is not
   limited by the nectar of the plant considered), leading to a plant biomass crash
   followed by insect decline.

   The graphical analysis provides another indication that oscillations are
   herbivory driven. On the one hand insect isoclines (or rather larva isoclines)
   are always positively sloped, and insects only grow when plant biomass is large
   enough (how large depends on insect's population size, due to intra-specific
   competition). Plant isoclines, on the other hand, can display a hump ([186]Fig.
   5B,C,D), and they grow (decrease) below (above) the hump. These two features of
   insect and plant isoclines are associated with limit cycles in classical
   victim-exploiter models [[187]27]. If there is no herbivory or another form of
   antagonism (e.g. competition) but only mutualism, the plant's isocline would be a
   positively sloped line, and plants would attain large populations in the presence
   of large insect populations, without cycles. However, mutualism is still
   essential for limit cycles: if mutualistic benefits are not large enough (gsa <
   yn), plant isoclines do not have a hump ([188]Fig. 5A) and oscillations are
   predicted to vanish. The effect of mutualism on stability is like the effect of
   enrichment on the stability in pure victim-exploiter models [[189]28], by
   allowing the plants to overcome the limits imposed by their intrinsic carrying
   capacity.

   There is a minor caveat regarding our graphical analysis: whereas a hump in the
   plant's isocline is a requisite for oscillations to evolve into limit cycles,
   this does not mean that isocline intersections at the left of the hump always
   lead to limit cycles ([190]Fig. 5C). To our best knowledge, this always happens
   only for quite specific conditions in pure victim-exploiter models [[191]40]. As
   long as we cannot prove by analytical means that intersection geometry determines
   local stability, the prediction of limit cycles remains a "rule of thumb", based
   on extrapolating our knowledge about other victim-exploiter models.

Classification of outcomes: mutualism or herbivory?

   Interactions can be classified according to the net effect of one species on the
   abundance (biomass, density) of another (but see other schemes [[192]41]). This
   classification scheme can be problematic in empirical contexts because reference
   baselines such as carrying capacities are usually not known [[193]42].

   Our PLA model illustrates the classification issue when non-equilibrium dynamics
   are generated endogenously, i.e. not by external perturbations. Since plants are
   facultative mutualists and insects are obligatory ones, one can say the outcome
   is net mutualism (+,+) or net herbivory (-,+), if the coexistence is stable, and
   the plant equilibrium ends up respectively above or below the carrying capacity
   [[194]33, [195]34]. If coexistence is under non-equilibrium conditions and plant
   oscillations are entirely below the carrying capacity (e.g. for large herbivory
   rates), the outcome is detrimental for plants and hence there is net herbivory
   (-,+); oscillations may in fact be considered irrelevant for this conclusion (or
   may further support the case of herbivory, read below). However, when the plant
   oscillation maximum is above carrying capacity and the minimum is below, like in
   [196]Fig. 3, could we say that the system alternates periodically between states
   of net mutualism and net herbivory? Here perhaps a time-based average over the
   cycle can help up us decide. The situation could be more complicated if plant
   oscillations lie entirely above the carrying capacity (see an example in [197]S1
   File): one can say that the net outcome is a mutualism due to enlarged plant
   biomasses, but the oscillations indicates that a victim-exploiter interaction
   exists. As we can see, deciding upon the net outcome require consideration of
   both equilibrium and dynamical aspects.

Factors that could cause dynamical transitions

Environmental factors.

   The parameters in our analyses can change due to external factors. One of the
   most important is temperature [[198]43]. It is well known, for example, that
   climate warming can reduce the number of days needed by larvae to complete their
   development [[199]44], making larvae maturation rates (g) higher. For insects
   that display Allee effects, a cooling of the environment will cause the sudden
   extinction of the insect and a catastrophic collapse of the mutualism, which
   cannot be simply reverted by warming. By retarding larva development into adults,
   cooling would increase the burden of herbivory over the benefits of pollination,
   making the system less stable by promoting oscillations. Flowering, pollination,
   herbivory, growth and mortality rates (e.g. s, a, b, r, m and n in equations
   [200]1) are also temperature-dependent and they can increase or decrease with
   warming depending on the thermal impacts on insect and plant metabolisms
   [[201]45]. This makes general predictions more difficult. However, we get the
   general picture that warming or cooling can change the balance between costs and
   benefits impacting the stability of the plant-insect association.

   Dynamical transitions can also be induced by changes in the chemical environment,
   often as a consequence of human activity. Some pesticides, for example, are
   hormone retarding agents [[202]46]. This means that their release can reduce
   maturation rates, altering the balance of the interaction towards more herbivory
   and less pollination and finally endangering pollination service [[203]47,
   [204]48]. In other cases, the chemical changes are initiated by the plants: in
   response to herbivory, many plants release predator attractants [[205]49], which
   can increase larval mortality (µ). If the insect does nothing but harm, this is
   always an advantage. If the insect is also a very effective pollinator, the abuse
   of this strategy can cost the plant important pollination services because a dead
   herbivore today is one less pollinator tomorrow.

   Another factor that can increase or decrease larvae maturation rates, is the
   level of nutrients present in the plant's vegetative tissue [[206]50, [207]51].
   On the one hand, the use of fertilizers rich in phosphorus could increase larvae
   maturation rates [[208]51]. On the other hand, under low protein consumption M.
   sexta larvae could decrease maturation rate, although M. sexta larvae can
   compensate this lack of proteins by increasing their herbivory levels (i.e.
   compensatory consumption) [[209]50]. Thus, different external factors related to
   plant nutrients could indirectly trigger different larvae maturation rates that
   will potentially modify the interaction dynamics.

Pollinator's diet breadth.

   An important factor that can affect the balance between mutualism and herbivory
   is the diet breadth of pollinators. Alternative food sources for the adults could
   lead to apparent competition [[210]52] mediated by pollination, as predicted for
   the interaction between D. wrigthii (Solanacea) and M. sexta (Sphingidae) in the
   presence of Agave palmieri (plant) [[211]6]: visitation of Agave by M. sexta does
   not affect the pollination benefits received by D. wrightii, but it increases
   oviposition rates on D. wrightii, increasing herbivory. As discussed before, such
   an increase in herbivory could explain why oscillations are more widespread when
   adult insects have alternative food sources (phi > 0) in our PLA model.

   Although we did not explore this with our model, the diet breadth of the larva
   could also have important consequences. In the empirical systems that inspired
   our model, the larva can have alternative hosts [[212]14], spreading the costs of
   herbivory over several species. The local extinction of such hosts could increase
   herbivory on the remaining ones, promoting unstable dynamics. To explore these
   issues properly, models like ours must be extended to consider larger community
   modules or networks, taking into account that there is a positive correlation
   between the diet breadths of larval and adult stages [[213]7].

   From the perspective of the plant, the lack of alternative pollinators could also
   lead to increased herbivory and loss of stability. The case of the tobacco plant
   (N. attenuata) and M. sexta is illustrative. These moths are nocturnal
   pollinators, and in response to herbivory by their larvae, the plants can change
   their phenology by opening flowers during the morning instead. Thus, oviposition
   and subsequent herbivory can be avoided, whereas pollination can still be
   performed by hummingbirds [[214]11]. Although hummingbirds are thought to be less
   reliable pollinators than moths for several reasons [[215]9], they are an
   alternative with negligible costs. Thus, a decline of hummingbird populations
   will render the herbivore avoidance strategy useless and plants would have no
   alternative but to be pollinated by insects with herbivorous larvae that promote
   oscillations.

Conclusions

   Many insect pollinators are herbivores during their larval phases. If pollination
   and herbivory targets the same plant (e.g. as between tobacco plants and
   hawkmoths), the overall outcome of the association depends on the balance between
   costs and benefits for the plant. As predicted by our plant-larva-adult (PLA)
   model, this balance is affected by changes in insect development: the faster
   larvae turns into adults the better for the plant and the interaction is more
   stable; the slower this development the poorer the outcome for the plant and the
   interaction is less stable (e.g. oscillations). Under plant-insect oscillations,
   this balance can be dynamically complex (e.g. periodic alternation between
   mutualism and antagonism). Since maturation rates play an essential role in long
   term stability, we predict important qualitative changes in the dynamics due to
   changes in environmental conditions, such as temperature and chemical compounds
   (e.g. toxins, hormones, plant nutrients). The stability of these mixed
   interactions can also be greatly affected by changes in the diet generalism of
   the pollinators.

Supporting Information

[216]S1 File. Supplement containing the appendices cited in the main text.

   [217]https://doi.org/10.1371/journal.pone.0117964.s001

   (PDF)

[218]S1 Fig. Detail of the b vs g parameter space for specialist pollinators in the PLA
model.

   The ellipse describes the joint variation of g and b taking place in the
   bifurcation diagram in [219]S2 Fig.

   [220]https://doi.org/10.1371/journal.pone.0117964.s002

   (EPS)

[221]S2 Fig. Bifurcation diagram for the PLA model.

   Parameters g and b vary along the elliptical path drawn in [222]S1 Fig, with
   reference for each quarter of a rotation. Solid (broken) lines represent stable
   (unstable) equilibria, black (white) circles represent limit cycle maxima and
   minima. The x = 1 line corresponds to the plant carrying capacity. HB[super]:
   super-critical and HB[sub]: sub-critical Hopf bifurcations, BP: branching point
   (transcritical bifurcation), LP: limit point (fold bifurcation).

   [223]https://doi.org/10.1371/journal.pone.0117964.s003

   (EPS)

[224]S3 Fig. Main configurations of the plant isocline.

   We only consider the O-K segment in the positive octant (hatched square). In A
   the isocline lies below the plant's carrying capacity (i.e. left of K), in B
   parts of the isocline lie above (i.e. right of K).

   [225]https://doi.org/10.1371/journal.pone.0117964.s004

   (EPS)

[226]S4 Fig. Shape of the plant's isocline.

   (A) As g increases and y, n decrease, points P and Q move closer to the diagonal
   asymptote (broken line), and the isocline eventually adopts the form of a
   mushroom. (B) As b increases, O, P, Q and the diagonal asymptote move towards the
   plant axis and the isocline is compressed vertically.

   [227]https://doi.org/10.1371/journal.pone.0117964.s005

   (EPS)

[228]S5 Fig. Main configurations of the larva isocline.

   The isocline consists of three black lines, but only the segment in the positive
   octant (hatched square) is biologically relevant. For A and B phi < n. For C and
   D phi > n. The green parabola p(x) is the numerator of the isocline and the
   circles indicate its roots, where x[0]: positive root. The red parabola q(x) is
   the denominator of the isocline, which has two roots x = 0 and x = x[v], both of
   which are also the vertical asymptotes of the isocline. The isocline also has an
   horizontal asymptote y[h]. The alternative in part D can be dismissed because it
   implies a detrimental effect of plants on insects.

   [229]https://doi.org/10.1371/journal.pone.0117964.s006

   (EPS)

[230]S6 Fig. Shape of the larva isocline.

   (A) For phi < n the larva isocline moves closer to the larva axis and becomes
   more shallow as g and b increase. (B) For phi > n the larva isocline becomes
   closer to the larva axis.

   [231]https://doi.org/10.1371/journal.pone.0117964.s007

   (EPS)

[232]S7 Fig. Plant oscillating above their carrying capacity in the PLA model.

   Blue:plant, green:larva, red:adult. The carrying capacity is indicated by the
   dotted line.

   [233]https://doi.org/10.1371/journal.pone.0117964.s008

   (EPS)

[234]S8 Fig. Oscillations in the PLA model started with different initial conditions
(*).

   The oscillations can dampen out (blue) or converge to a limit cycle (red).

   [235]https://doi.org/10.1371/journal.pone.0117964.s009

   (EPS)

[236]S9 Fig. Interaction dynamics when flowers are explicitly considered.

   Blue:plant, green:larva, red:adult, black:flowers. The dotted line indicates the
   plant's carryng capacity.

   [237]https://doi.org/10.1371/journal.pone.0117964.s010

   (EPS)

Acknowledgments

   We thank Rampal Etienne for the discussions that inspired us to write this
   article. We thank the comments and suggestions from our colleagues of the Centre
   for Biodiversity Theory and Modelling in Moulis, France, the Centre for
   Australian National Biodiversity and Research at CSIRO in Canberra, Australia,
   and two anonymous reviewers.

Author Contributions

   Conceived and designed the experiments: TAR FEV. Performed the experiments: TAR.
   Analyzed the data: TAR. Contributed reagents/materials/analysis tools: TAR. Wrote
   the paper: TAR FEV.

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